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Volume -2, Issue-6, November - December 2016, Page No. 09 -14

e

9

ISSN: 2455 - 1597

Similarity Measures of Intuitionistic Fuzzy Sets Dr. Surendra Prasad Jena, Reader in Mathematics

S.B. Women’s College, Cuttack E-mail: sp.jena08@gmail.com Abstract

In this paper, we propose two similarity measures for measuring the degree of similarity between intuitionistic fuzzy sets.

Keywords: Fuzzy Sets, Intuitionistic Fuzzy Sets, Similarity Measure. 1. Introduction

The concept of fuzzy set was introduced by Zadeh in his classical paper ([6]) in 1965. In 1975, in another direction,

Atanassov ([1]) introduced the concept of intuitionistic fuzzy set. Many measures of similarity between fuzzy sets have

been proposed in ([7], [3]). In ([5]), S.M. Chen, proposed two methods for measures of similarity between vague sets.

Also, using vector approach, Chen ([2], [4]) defined a measure of similarity between two fuzzy sets.

In the present paper, we propose some methods for measures of similarity between intuitionistic fuzzy sets of same nature

of S.M. Chen’s ([5]).

Intuitionistic Fuzzy Sets

Let a set X be fixed. An IFS Ai in X consists the performance value Pij, for fixed i and j = 1, 2, 3, … having the

form

( )

( )

{

x

,

x

,

x

:

x

X

}

A

i j A j A j j

i

i

ν

µ

=

{

µ

ij

,

ν

ij

:

j

=

1

,

2

,

3

,

...

}

=

.

Where the performance value

P

ij

=

µ

ij

,

ν

ij

,

j

=

1

,

2

,

,

...

and the function

µ

ij

:

X

[ ]

0

,

1

and

ν

ij

:

X

[ ]

0

,

1

defines the degree of membership and degree of non-membership respectively of the element

x

j

X

to the set Ai, which

is a subset of X and for every

x

j

X

,

0

µ

ij

+

ν

ij

1

.

It is clear that, A performance value

P

ij

=

µ

ij

,

ν

ij consists of the membership value

µ

ij and the

non-membership value

ν

ij. Clearly the hesitation part is

ij ij ij

=

1

µ

ν

π

we use the following notation.

( )

P

ij

=

µ

ij

,

ν

( )

P

ij

=

ν

ij

and

π

( )

P

ij

=

π

ij

(2)

Pag

e

10

Pag

e

10

Pag

e

10

Pag

e

10

Pag

e

10

Pag

e

10

Pag

e

10

Pag

e

10

Pag

e

10

Pag

e

10

Pag

e

10

Pag

e

10

Pag

e

10

Pag

e

10

Pag

e

10

Pag

e

10

Pag

e

10

Pag

e

10

Pag

e

10

Pag

e

10

Pag

e

10

Example 1.1: Consider two IFSs A1 and A2 of

X

=

{

x

1

,

x

2

,

x

3

,

x

4

}

given by the following table

X

1

1 A

A

,

ν

µ

2

2 A

A

,

ν

µ

x1 .60, .30 .65, .25

x2 .70, .20 .75, .20

x3 .80, .12 .31, .44

x4 .90, .02 .52, .32

That is

(

)

(

)

(

)

(

)

{

1 2 3 4

}

1

.

60

,

.

30

x

,

.

70

,

.

20

x

,

.

80

,

.

12

x

,

.

90

,

.

02

x

A

=

(

)

(

)

(

)

(

)

{

1 2 3 4

}

2

.

65

,

.

25

x

,

.

75

,

.

20

x

,

.

31

,

.

44

x

,

.

52

,

.

32

x

A

=

that is the performance values are

(

) (

11 11

)

(

A

( )

1 A

( )

1

)

11

.

60

,

.

30

,

x

,

x

P

1

1

ν

µ

=

ν

µ

=

=

(

) (

12 12

)

(

A

( )

2 A

( )

2

)

12

.

70

,

.

20

,

x

,

x

P

1

1

ν

µ

=

ν

µ

=

=

(

) (

13 13

)

(

A

( )

3 A

( )

3

)

13

.

80

,

.

12

,

x

,

x

P

1

1

ν

µ

=

ν

µ

=

=

(

) (

14 14

)

(

A

( )

4 A

( )

4

)

14

.

90

,

.

02

,

x

,

x

P

=

=

µ

ν

=

µ

1

ν

1

(

) (

21 21

)

(

A

( )

1 A

( )

1

)

21

.

65

,

.

25

,

x

,

x

P

2

2

ν

µ

=

ν

µ

=

=

(

) (

22 22

)

(

A

( )

2 A

( )

2

)

22

.

75

,

.

20

,

x

,

x

P

2

2

ν

µ

=

ν

µ

=

=

(

) (

23 23

)

(

A

( )

3 A

( )

3

)

23

.

31

,

.

44

,

x

,

x

P

2

2

ν

µ

=

ν

µ

=

=

(

) (

24 24

)

(

A

( )

4 A

( )

4

)

24

.

52

,

.

32

,

x

,

x

P

2

2

ν

µ

=

ν

µ

=

=

2. Similarity Measure:

Let

x

=

(

µ

x

,

ν

x

)

be a performance value, where

µ

x

[ ]

0

,

1

and

ν

x

[ ]

0

,

1

,

0

µ

x

+

ν

x

1

. Then the measure of ‘x’ can be evaluated by a measure function ‘m’ shown as follows:

(2.1)

m

( )

x

=

2

µ

x

+

π

x

1

where

π

x

=

1

µ

x

ν

x be the hesitation part and

m

( )

x

[

1

,

1

]

(3)

e

11

e

11

e

11

e

11

e

11

e

11

e

11

e

11

e

11

e

11

e

11

e

11

e

11

e

11

e

11

e

11

e

11

e

11

e

11

e

11

e

11

(2.2)

( )

( ) ( )

2

y

m

x

m

1

y

.

x

S

=

where

m

( )

x

=

2

µ

x

+

π

x

1

and

m

( )

y

=

2

µ

y

+

π

y

1

Case-I If the performance values x = (1, 0) and y = (0, 1), then we can see that

( )

x

2

.

1

0

1

1

and

m

( )

y

2

0

0

1

1

m

=

+

=

=

+

=

By applying equation 2.2, the degree of similarity between x and y can be evaluated and is equal to

( )

( )

1

1

0

2

1

1

1

y

.

x

S

=

=

=

.

Case-II If the performance values x = (1, 0) and y = (1, 0), then we can see that

( )

x

1

and

m

( )

y

1

m

=

=

.

So, degree of similarity between x and y can be evaluated and is equal to

( )

1

0

1

2

1

1

1

y

.

x

S

=

=

=

.

Case-III If the performance values x = (0, 1) and y = (0, 1), then we can see that

( )

x

1

and

m

( )

y

1

m

=

=

.

So, the degree of similarity between x and y can be evaluated and is equal to

( )

1

0

1

2

1

1

1

y

.

x

S

=

+

=

=

.

Case-IV If the performance values x = (0, 1) and y = (1, 0), then we can see that

( )

x

1

and

m

( )

y

1

m

=

=

.

So, the degree of similarity between x and y can be evaluated and is equal to

( )

1

1

0

2

1

1

1

y

.

x

S

=

=

=

.

It is obvious that if x and y are identical performance value, then m(x) = m(y) and the degree of similarity between the

performance values x and y is equal to 1.

Let A1 and A2 be two IFSs in

X

=

{

x

1

,

x

2

,

x

3

...

x

n

}

. Then

( )

( )

(

)

(

( )

( )

)

(

( )

( )

)

{

A 1 A 1 1 A 2 A 2 2 A n A n n

}

1

x

,

x

x

,

x

,

x

x

,

...

x

,

x

x

A

1 1

1 1

1

1

ν

µ

ν

µ

ν

µ

=

( )

( )

(

)

(

( )

( )

)

(

( )

( )

)

{

x

,

x

x

,

x

,

x

x

,

...

x

,

x

x

,

}

A

2 A 1 A 1 1 A 2 A 2 2 A n A n n

2 2

2 2

2

2

ν

µ

ν

µ

ν

(4)

Pag e

12

Pag e

12

Pag e

12

Pag e

12

Pag e

12

Pag e

12

Pag e

12

Pag e

12

Pag e

12

Pag e

12

Pag e

12

Pag e

12

Pag e

12

Pag e

12

Pag e

12

Pag e

12

Pag e

12

Pag e

12

Pag e

12

Pag e

12

Pag e

12

So that

( )

( )

(

x

,

x

) (

,

)

,

i

1

,

2

,

...

n

P

i 1 i 1 1

1 i A i A A

A i

1

=

µ

ν

=

µ

ν

=

and

P

(

( )

x

,

( )

x

) (

,

)

,

i

1

,

2

,

...

n

i 2 i 2 2

2 i A i A A

A i

2

=

µ

ν

=

µ

ν

=

be the performance values of ‘xi’ in the IFSs A1 and A2 respectively, then applying equation 2.1, we can see that

(2.3)

( )

( )

( )

( )

P

2

( )

x

( )

x

1

,

1

i

n

m

and

1

x

x

2

P

m

i A i A i 2 i A i A i 1 2 2 1 1

π

+

µ

=

π

+

µ

=

and the degree of similarity between the IFSs A1 and A2 can be evaluated by the function T, as

(2.4)

(

)

(

1 2

)

n

1 i 2

1

S

A

,

A

n

1

A

,

A

T

=

( ) ( )





=

2

P

m

P

m

1

n

1

n 1i 2i

1 i

where

T

(

A

1

,

A

2

)

[ ]

0

,

1

. The larger value of

T

(

A

1

,

A

2

)

, the more the similarity between the IFSs A1 and A2.

Example 2.1: Let

X

=

{

x

1

,

x

2

,

x

3

,

x

4

}

be a fixed set and

(

)

(

)

(

)

(

)

{

1 2 3 4

}

1

.

60

,

.

30

x

,

.

70

,

.

20

x

,

.

80

,

.

12

x

,

.

90

,

.

02

x

A

=

(

)

(

)

(

)

(

)

{

1 2 3 4

}

2

.

65

,

.

25

x

,

.

75

,

.

20

x

,

.

31

,

.

44

x

,

.

52

,

.

32

x

A

=

where

(

) (

11 11

)

(

A

( )

1 A

( )

1

)

11

.

60

,

.

30

,

x

,

x

P

1 1

ν

µ

=

ν

µ

=

=

(

) (

12 12

)

(

A

( )

2 A

( )

2

)

12

.

70

,

.

20

,

x

,

x

P

1 1

ν

µ

=

ν

µ

=

=

(

) (

13 13

)

(

A

( )

3 A

( )

3

)

13

.

80

,

.

12

,

x

,

x

P

1 1

ν

µ

=

ν

µ

=

=

(

) (

14 14

)

(

A

( )

4 A

( )

4

)

14

.

90

,

.

02

,

x

,

x

P

1 1

ν

µ

=

ν

µ

=

=

(

) (

21 21

)

(

A

( )

1 A

( )

1

)

21

.

65

,

.

25

,

x

,

x

P

2 2

ν

µ

=

ν

µ

=

=

(

) (

22 22

)

(

A

( )

2 A

( )

2

)

22

.

75

,

.

20

,

x

,

x

P

2 2

ν

µ

=

ν

µ

=

=

(

) (

23 23

)

(

A

( )

3 A

( )

3

)

23

.

31

,

.

44

,

x

,

x

P

2 2

ν

µ

=

ν

µ

=

=

(

) (

24 24

)

(

A

( )

4 A

( )

4

)

24

.

52

,

.

32

,

x

,

x

P

2 2

ν

µ

=

ν

µ

=

=

Now applying equation 2.1, we can get

( )

P

2

( )

x

( )

x

1

2

.

60

.

10

1

.

30

m

11 A 1 A 1

1

1

+

π

=

×

+

=

(5)

e

13

e

13

e

13

e

13

e

13

e

13

e

13

e

13

e

13

e

13

e

13

e

13

e

13

e

13

e

13

e

13

e

13

e

13

e

13

e

13

e

13

( )

P

2

( )

x

( )

x

1

2

.

70

.

10

1

.

50

m

12 A 2 A 2

1

1

+

π

=

×

+

=

µ

=

( )

P

2

( )

x

( )

x

1

2

.

80

.

08

1

.

68

m

13 A 3 A 3

1

1

+

π

=

×

+

=

µ

=

( )

P

2

( )

x

( )

x

1

2

.

90

.

08

1

.

88

m

14 A 4 A 4

1

1

+

π

=

×

+

=

µ

=

( )

P

2

( )

x

( )

x

1

2

.

65

.

10

1

.

40

m

21 A 1 A 1

2

2

+

π

=

×

+

=

µ

=

( )

P

2

( )

x

( )

x

1

2

.

75

.

05

1

.

55

m

22 A 2 A 2

2

2

+

π

=

×

+

=

µ

=

( )

P

2

( )

x

( )

x

1

2

.

31

.

25

1

.

13

m

23 A 3 A 3

2

2

+

π

=

×

+

=

µ

=

( )

P

2

( )

x

( )

x

1

2

.

52

.

16

1

.

20

m

24 A 4 A 4

2

2

+

π

=

×

+

=

µ

=

By applying equation 2.4, the degree of similarity between A1 and A2 can be evaluated by

(

)

n

(

1 2

)

1 i 2

1

S

A

,

A

n

1

A

,

A

T

=

( ) ( )





=

2

P

m

P

m

1

4

1

4 1i 2i

1 i

[

.

95

.

975

.

595

.

66

]

.

795

4

1

+

+

+

=

=

which indicate that the degree of similarity between the IFSs A1 and A2 be .795.

In the following, we present a weighted similarity measure between IFSs A1 and A2. Let A1 and A2 be two IFSs

{

x

1

,

x

2

,

x

3

...

x

n

}

X

=

. Then

( )

( )

(

)

1i i

n

1 i i i A i A n

1 i

1

x

,

x

x

P

x

A

1

1

= =

=

ν

µ

=

and

( )

( )

(

)

2i i

n

1 i i i A i A n

1 i

2

x

,

x

x

P

x

A

2 2

= =

=

ν

µ

=

Assume that, the weight of the elements xi in X is wi respectively, where

w

i

[ ]

0

,

1

and

1

i

n

. Then the degree of

similarity between IFSs A1 and A2 can be evaluated by the weighting function ‘W’

(2.5)

(

)

( ) ( )

i

n

1 i i 2 i

1 *

i n

1 i 2

1

w

2

P

m

P

m

1

w

A

,

A

W

=

=





=

where

W

(

A

1

,

A

2

)

[ ]

0

,

1

. Then the larger value of

W

(

A

1

,

A

2

)

, the more similarity between the IFSs A1 and A2.

Example 2.2: Consider the previous example we have

( )

P

.

30

(6)

Pag

e

14

Pag

e

14

Pag

e

14

Pag

e

14

Pag

e

14

Pag

e

14

Pag

e

14

Pag

e

14

Pag

e

14

Pag

e

14

Pag

e

14

Pag

e

14

Pag

e

14

Pag

e

14

Pag

e

14

Pag

e

14

Pag

e

14

Pag

e

14

Pag

e

14

Pag

e

14

Pag

e

14

( )

P

.

50

m

12

=

m

( )

P

22

=

.

55

( )

P

.

68

m

13

=

m

( )

P

23

=

.

13

( )

P

.

88

m

14

=

m

( )

P

24

=

.

20

Assume that the weights of x1, x2, x3 and x4 be .5, .8, 1.0 and 0.7 respectively. Then applying equation 2.5, the degree of

similarity between IFSs A1 and A2 can be evaluated by the weighting function W as

(

)

( ) ( )

n i

1 i i 2 i

1 *

i n

1 i 2

1

w

2

P

m

P

m

1

w

A

,

A

W

=

=





=

7

.

0

.

1

8

.

5

.

66

.

7

.

595

.

0

.

1

975

.

8

.

95

.

5

.

+

+

+

×

+

×

+

×

+

×

=

3

462

.

595

.

78

.

475

.

+

+

+

=

6

770

.

=

771

.

=

3. Conclusion

Although many similarity measure have been proposed in the literature for measuring the degree of similarity between

fuzzy sets. In this section we propose similarity measures for measuring the degree of similarity between IFSs. The

proposed measures can provide a useful way to deal with the similarity measures of IFSs.

4. Reference:

[1]. Atanassov, K., Intuitionistic fuzzy sets, Fuzzy Sets and Systems. 20 (1986), 87-96.

[2]. Chen, S.M., A new approach to handling fuzzy decision making problems, IEEE Trans. on Sys. Man. Cyber. SMC –

18 (6) (Nov/Dec, 1988), 1311-1347.

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